Why does part of my mesh have a texture even though its UVs are off the edge of the image in the UV Editor?
The texture has repeated itself
A texture is by default tiled, you can see the actual repetition by enabling the Repeat Image checkbox in the View tab of the UV editor.
This is usually very useful when you are using seamless textures.

If you want to avoid this behaviour you have to explicitly clamp (clip) your UV texture coordinates.